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Optimal decay for a wave-heat system with Coleman-Gurtin thermal law (2101.11397v3)
Published 27 Jan 2021 in math.AP, math.FA, and math.OC
Abstract: We study the long-term behaviour of solutions to a one-dimensional coupled wave-heat system with Coleman-Gurtin thermal law. Our approach is based on the asymptotic theory of $C_0$-semigroups and recent results developed for coupled control systems. As our main results, we represent the system as a feedback interconnection between the wave part and the Coleman-Gurtin part and we show that the associated semigroup in the history framework of Dafermos is polynomially stable with optimal decay rate $t{-2}$ as $t\to\infty$. In particular, we obtain a sharp estimate for the rate of energy decay of classical solutions to the problem.